= Solution
Let $S_n=\sum_{i\leq n}a_i$ and $T_n=\sum_{i\leq n}|a_i|$. Conditional convergence gives $S_n\to S$ and $T_n\to\infty$. Since $P_n=T_n+S_n$ and $N_n=T_n-S_n$,
$$
\frac{P_n}{N_n}=\frac{1+S_n/T_n}{1-S_n/T_n}\longrightarrow1.
$$
The alternating-series test says that $b_n\downarrow0$ implies convergence of $\sum(-1)^nb_n$. The hypothesis implies $b_n/b_{n+1}>1$ eventually, so $b_n$ is eventually decreasing. For some $c>0$, eventually $b_n/b_{n+1}\geq1+c/n$; the divergent product of these factors forces $b_n\to0$. The test now applies.
Back to article page
